Related papers: Solving effectively some families of Thue Diophant…

We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in…

A rational Diophantine $m$-tuple is a set $\{a_1,\ldots,a_m\}$ of distinct nonzero rational numbers such that $a_i a_j+1$ is a square for all $1\leq i < j\leq m$. Similarly, we may ask when $a_ia_j+1$ is a $k$-th power. Here, we study the…

In this paper we deal with a non-linear Diophantine equation which arises from the determinant computation of an integer matrix. We show how to find a solution, when it exists. We define an equivalence relation and show how the set of all…

We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d.…

Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every…

Consider an irreducible bilinear form $f(x_1,x_2;y_1,y_2)$ with integer coefficients. We derive an upper bound for the number of integer points $(\mathbf{x},\mathbf{y})\in\mathbb{P}^1\times\mathbb{P}^1$ inside a box satisfying the equation…

We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$,…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…

In this paper we solve the Diophantine equation $\binom{m}{l}-\binom{n}{k}=d$ (where m,n are positive integers unknowns) when (k,l)=(3,6) for various values of d and when (k,l)=(8,2) and d=1. As a byproduct of our results we will obtain…

Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive…

We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…

Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence.…

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot…

We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $\abs{Q(x)-\xi}< \epsilon$ for a generic ternary form $Q$ is $\abs{x}\ll \epsilon^{-1}$. We also establish…

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative…

The seminal work of Kurzweil (1955) provides for any fixed badly approximable $\alpha$ and monotonically decreasing $\psi$ a Khintchine-type statement on the set of the inhomogeneous real parameters $\gamma$ for which $\lVert n \alpha +…

Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in…

We study solvability of the Diophantine equation \begin{equation*} \frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in integers $n, k, a_{1},\ldots, a_{k}$ satisfying the conditions $k\geq 2$ and $a_{i}<a_{i+1}$ for…

Let $Q$ be an infinite subset of $\mathbb{Z}$, let $\Psi: \mathbb{Z} \rightarrow [0,\infty)$ be positive on $Q$, and let $\theta \in \mathbb{R}$. Define $$ E(Q,\Psi,\theta) = \{ x \in \mathbb{R} : \| q x - \theta \| \leq \Psi(q) \text{ for…

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were…